{
  "abstract": "For functions that are convex of higher order, classical extremalities identify Gaussian quadrature as a lower bound and Gauss--Lobatto quadrature as an upper bound for the integral. We sharpen this bracket in every odd order: if $f$ is $(2n-1)$-convex on $[-1,1]$, then the exact integral $I[f]:=\\int_{-1}^{1}f(x)\\,dx$ lies not merely between the $n$-point Gauss--Legendre rule $G_n[f]$ and the $(n+1)$-point Gauss--Lobatto rule $L_{n+1}[f]$, but already between $G_n[f]$ and their midpoint. This yields the certified estimator $Q_n[f]:=\\frac{3}{4}G_n[f]+\\frac{1}{4}L_{n+1}[f]$, with $|I[f]-Q_n[f]|\\leqslant\\frac{1}{4}|L_{n+1}[f]-G_n[f]|$, and the constant $\\frac{1}{4}$ is sharp. We also record a complementary phenomenon in even order: for $2n$-convex functions the Gauss--Radau endpoint rules still bracket the integral, but the midpoint of the two Radau rules cannot yield a one-sided refinement. The obstruction is explained by an odd, sign-changing Radau quadrature kernel. Finally, we distinguish numerical certification from ordinary high-order approximation: we include concrete examples from spline, moment and statistical models and report experiments for truncated powers and near-pole Stieltjes kernels, where the shape certificate remains meaningful when classical smoothness-based error constants are unavailable or severely pessimistic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34918",
  "author_total": 2,
  "authors": [
    {
      "name": "Tomasz Kania"
    },
    {
      "name": "Szymon Wąsowicz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fypbrs9jxahpb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34918",
  "fulltext_url": null,
  "id": "fypbrs9jxahpb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34918",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.34918",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fypbrs9jxahpb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34918",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34918"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Refined Gauss--Lobatto bounds for odd-order convexity",
  "title_text": "Refined Gauss--Lobatto bounds for odd-order convexity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/fypbrs9jxahpb",
  "venue": "arXiv",
  "versions": []
}